Egg mayonnaise is the same bound filling as egg mayo carried under its full name, and the register is the point. The contraction belongs to the counter queue and the meal deal; the full word belongs to the menu card, the sandwich-round list, the afternoon tea where things are written out properly. Nothing in the bowl changes, but the name signals a context where the sandwich is presented rather than grabbed: trimmed, cut small, plated, treated as a thing made with care rather than a default reached for without thinking. That is worth saying plainly, because the difference between egg mayo and egg mayonnaise is not the egg. It is where the sandwich is being eaten and how much attention the making of it expects.
The craft is the bind, executed with the tidiness the formal name implies. Hard-cooked egg is chopped and held with just enough mayonnaise to cohere, seasoned with salt and white pepper through the filling rather than on the bread, and the ratio is the whole discipline: enough binder to bring it together, not so much that it slumps. In the presented version the egg is often mashed finer for a smooth, even face that cuts cleanly, the bread is soft white, buttered to the edges to seal the crumb, the crusts taken off and the sandwich cut into fingers or triangles small enough to eat without a plate. It is built close to serving and pressed gently so the filling holds a clean line when it is cut, because a formal sandwich that spills its filling defeats the register it was made for.
The variations are the rest of the bound-egg family, each the same base under a different addition or a different name. Egg mayo is this exact filling in the colloquial, grab-and-go register; egg and cress and egg and watercress add a peppery green; egg and chive an allium edge; egg and salad cream a sharper, thinner binder. Each deserves its own article rather than being crowded in here.